Solution
= Solution
Parametrize the unit circle by a loop $\gamma:[0,2\pi]\to\mathbb C^*$. Part (d) and $f(\gamma(0))=f(\gamma(2\pi))$ imply
$$
\exp\left(-\int_{S^1}\beta\right)=1.
$$
The kernel of the <complex exponential function> is $2\pi i\mathbb Z$, so
$$
\int_{S^1}\beta=2\pi i k
$$
for some $k\in\mathbb Z$. This integer is the <winding number> of $f|_{S^1}$.